How can we rigorously quantify Monte Carlo error and assess convergence in modern MCMC methods such as the No-U-Turn Sampler?
This question motivates new joint work with Victor de la Peña (Columbia), which introduces a decoupling-based perspective for Markov chains with direct applications to MCMC.
Core idea.
Essentially all MCMC algorithms — from Gibbs samplers to Hamiltonian Monte Carlo and NUTS — can be written as deterministic update maps driven by i.i.d. auxiliary random variables. While this representation is well known, we show that it can be exploited in a systematic and largely unexplored way.
The paper introduces a tangent-decoupling framework for Markov chains. Alongside the original chain, we construct a companion process by re-running the same update map along the realized trajectory, but with fresh auxiliary randomness injected at each step. The resulting tangent-decoupled sequence
- is straightforward to simulate in tandem with the original chain,
- is conditionally independent given the realized backbone trajectory, and
- is generally non-Markovian, yet remains ergodically correct.
Main results.
Two theoretical consequences of this construction are particularly striking.
First, we establish almost sure consistency of empirical averages computed from the tangent-decoupled sequence, despite its lack of Markovian structure. Under mild conditions, these averages converge to the correct target expectation, yielding a structurally simple and broadly applicable estimator.
Second, we prove a sharp, nonasymptotic variance inequality: for any square-integrable observable and any finite sample size, the variance of the standard MCMC estimator is bounded above by twice the variance of the tangent-decoupled estimator. This bound is assumption-free: it does not rely on reversibility, spectral gaps, or mixing-time arguments; and leads directly to principled uncertainty quantification for MCMC output.
Why the connection is interesting mathematically
Decoupling theory is a classical tool in probability, with deep connections to martingale theory, empirical processes, and the study of dependent random variables. At its core, decoupling provides a principled way to compare dependent processes with suitably constructed independent counterparts.
What is novel here is that this abstract theory is brought into direct contact with the concrete structure of Markov chains. By identifying a natural tangent-decoupled companion process, the work shows how general decoupling principles can be realized in an explicit, algorithmically meaningful setting.
Beyond the specific results, this perspective suggests a new way to think about convergence diagnostics and error assessment in MCMC algorithms, yielding tools for error assessment that remain valid even when classical Markov-chain assumptions are unavailable or hard to verify.
I am very grateful to Victor for this collaboration. The project was sparked by a question he raised after an applied probability seminar I gave at Columbia, a reminder of the well-known adage that good questions are often the engine of good mathematics, and highlighting the role of seminars not only as venues for dissemination, but as catalysts for new ideas and collaborations.
The full paper is available here: https://arxiv.org/abs/2512.19351